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Acceleration Analysis of 3-RPS Parallel Manipulators by Means of Screw Theory

J. Gallardo, H. Orozco, J.M. Rico, C.R. Aguilar and L. Pérez Department of Mechanical Engineering, Instituto Tecnológico de Celaya,

FIMEE, Universidad de Guanajuato, México

1. Introduction

According to the notation proposed by the International Federation for the Theory of Mechanisms and Machines IFToMM (Ionescu, 2003); a parallel manipulator is a mechanism where the motion of the end-effector, namely the moving or movable platform, is controlled by means of at least two kinematic chains. If each kinematic chain, also known popularly as limb or leg, has a single active joint, then the mechanism is called a fully-parallel mechanism, in which clearly the nominal degree of freedom equates the number of limbs. Tire-testing machines (Gough & Whitehall, 1962) and flight simulators (Stewart, 1965), appear to be the first transcendental applications of these complex mechanisms. Parallel manipulators, and in general mechanisms with parallel kinematic architectures, due to benefits --over their serial counterparts-- such as higher stiffness and accuracy, have found interesting applications such as walking machines, pointing devices, multi-axis machine tools, micro manipulators, and so on. The pioneering contributions of Gough and Stewart, mainly the theoretical paper of Stewart (1965), influenced strongly the development of parallel manipulators giving birth to an intensive research field. In that way, recently several parallel mechanisms for industrial purposes have been constructed using the, now, classical hexapod as a base mechanism: Octahedral Hexapod HOH-600 (Ingersoll), HEXAPODE CMW 300 (CMW), Cosmo Center PM-600 (Okuma), F-200i (FANUC) and so on. On the other hand one cannot ignore that this kind of parallel kinematic structures have a limited and complex-shaped workspace. Furthermore, their rotation and position capabilities are highly coupled and therefore the control and calibration of them are rather complicated. It is well known that many industrial applications do not require the six degrees of freedom of a parallel manipulator. Thus in order to simplify the kinematics, mechanical assembly and control of parallel manipulators, an interesting trend is the development of the so called defective parallel manipulators, in other words, spatial parallel manipulators with fewer than six degrees of freedom. Special mention deserves the Delta robot, invented by Clavel (1991); which proved that parallel robotic manipulators are an excellent option for industrial applications where the accuracy and stiffness are fundamental characteristics. Consider for instance that the Adept Quattro robot, an application of the Delta robot, developed by Francois Pierrot in collaboration with Fatronik (Int. patent appl. WO/2006/087399), has a

Source: Parallel Manipulators, New Developments, Book edited by: Jee-Hwan Ryu, ISBN 978-3-902613-20-2, pp. 498, April 2008, I-Tech Education and Publishing, Vienna, Austria

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Parallel Manipulators, New Developments

316

2.0 kilograms payload capacity and can execute 4 cycles per second. The Adept Quattro robot is considered at this moment the industry's fastest pick-and-place robot. Defective parallel manipulators can be classified in two main groups: Purely translational (Romdhane et al, 2002; Parenti-Castelli et al, 2000; Carricato & Parenti-Castelli, 2003; Di Gregorio & Parenti-Castelli, 2002; Ji & Wu, 2003; Kong & Gosselin, 2004a; Kong & Gosselin, 2002) or purely spherical (Alizade et al, 1994; Di Gregorio, 2002; Gosselin & Angeles, 1989; Kong & Gosselin, 2004b; Liu & Gao 2000). A third class is composed by parallel manipulators in which the moving platform can undergo mixed motions (Parenti-Castelli & Innocenti, 1992; Gallardo-Alvarado et al, 2006; Gallardo-Alvarado et al, 2007). The 3-RPS, Revolute + Prismatic +Spherical, parallel manipulator belongs to the last class and is perhaps the most studied type of defective parallel manipulator. The 3-RPS parallel manipulator was introduced by Hunt (1983) and has been the motive of an exhaustive research field where a great number of contributions, approaching a wide range of topics, kinematic and dynamic analyses, synthesis, singularity analysis, extensions to hyper-redundant manipulators, etc; have been reported in the literature, see for instance Lee & Shah (1987), Kim & Tsai (2003), Liu & Cheng (2004), Lu & Leinonen (2005). In particular, screw theory has been proved to be an efficient mathematical resource for determining the kinematic characteristics of 3-RPS parallel manipulators, see for instance Fang & Huang (1997), Huang and his co-workers (1996, 2000, 2001, 2002); including the instantaneous motion analysis of the mechanism at the level of velocity analysis (Agrawal, 1991). This paper addresses the kinematics of 3-RPS parallel manipulators, including position, velocity and acceleration analyses. Firstly the forward position analysis is carried out in analytic form solution using the Sylvester dialytic elimination method. Secondly the velocity and acceleration analyses are approached by means of the theory of screws. To this end, the velocity and reduced acceleration states of the moving platform, with respect to the fixed platform, are written in screw form through each one of the limbs of the mechanism. Finally, the systematic application of the Klein form to these expressions allows obtaining simple and compact expressions for computing the velocity and acceleration analyses. A case study is included.

2. Description of the mechanism

A 3-RPS parallel manipulator, see Fig. 1, is a mechanism where the moving platform is connected to the fixed platform by means of three extendible limbs. Each limb is composed by a lower body and an upper body connected each other by means of an active prismatic joint. The moving platform is connected at the upper bodies via three distinct spherical joints while the lower bodies are connected to the fixed platform by means of three distinct revolute joints. An effective general formula for determining the degrees of freedom of closed chains still in our days is an open problem. An exhaustive review of formulae addressing this topic is reported in Gogu (2005). Regarding to the existing methods of computation, these formulae are valid under specific conducted considerations. For the parallel manipulator at hand, the mobility is determined using the well-known Kutzbach-Grübler formula

∑+−−==

j

1iif1)j6(nF (1)

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Fig. 1. The 3-RPS parallel manipulator and its geometric scheme.

Where n is the number of links, j is the number of kinematic pairs and if is the number of

freedoms of the i-th pair. Thus, taking into account that for the mechanism at hand n=8, j=9

and 15ifj

1i

=∑=

; then the degrees of freedom of it are equal to 3, an expected result.

2. Position analysis

In this section the forward finite kinematics of the 3-RPS parallel manipulator is approached using analytic procedures. The inverse position analysis is considered here a trivial task and therefore it is omitted. The geometric scheme of the spatial mechanism is shown in the right side of Fig. 1.

Accordingly with this figure; iB , iq and iP denotes, respectively, the nominal position of

the revolute joint, the length of the limb and the center of the spherical joint in the same

limb. While iu denotes the direction of the axis associated to the revolute joint. On the other

hand mna represents the distance between the centers of two spherical joints.

In this work, the forward position analysis of the 3-RPS parallel manipulator consists of finding the pose, position and orientation, of the moving platform with respect to the fixed

platform given the three limb lengths or generalized coordinates iq of the parallel

manipulator. To this end, it is necessary to compute the coordinates of the three spherical joints expressed in the reference frame XYZ. When the limbs of the parallel manipulator are locked, the mechanism becomes a 3-RS structure. In order to simplify the analysis, the reference frame XYZ, attached at the fixed

platform, is chosen in such a way that the points iB lie on the XZ plane. Under this

consideration the axes of the revolute joints are coplanar and three constraints are imposed by these joints as follows

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( ) { }1,2,3i0iuiBiP ∈=•− (2)

where the dot denotes the usual inner product operation of the three dimensional vectorial algebra. It is worth to mention that expressions (2) were not considered, in the form derived, by Tsai (1999), and therefore the analysis reported in that contribution requires a particular arrangement of the positions of the revolute joints over the fixed platform accordingly to the reference frame XYZ. Furthermore, clearly expressions (2) are applicable not only to tangential 3-RPS parallel manipulators, like the mechanism of Fig. 1, but also to the so-called concurrent 3-RPS parallel manipulators. On the other hand, clearly the limb lengths are restricted to

( ) ( ) { }1,2,3i2iqiBiPiBiP ∈=−•− (3)

Finally, three compatibility constraints can be obtained as follows

( ) ( )( ) ( )( ) ( ) 2

12a2P1P2P1P

213a3P1P3P1P

223a3P2P3P2P

=−•−=−•−=−•−

(4)

Expressions (2)-(4) form a system of nine equations in the nine

unknowns{ }333222111 Z,Y,X,Z,Y,X,Z,Y,X . In what follows, expressions (2-4) are

reduced systematically into a highly non linear system of three equations in three unknowns. Afterwards, a sixteenth-order polynomial in one unknown is derived using the Sylvester dialytic elimination method. It follows from Eqs. (2) that

( ) { }1,2,3iiZfiX ∈= (5)

On the other hand with the substitution of (5) into expressions (3), the reduction of terms leads to

{ }1,2,3iip2iY ∈= (6)

where ip are second-degree polynomials in iZ . Finally, the substitution of Eqs. (6) into Eqs.

(4) results in the following highly non-linear system of three equations in the three

unknowns 1Z , 2Z and 3Z

0eZeZeZZeZZeZZeZeZe

0dZdZdZZdZZdZZdZdZd

0cZcZcZZcZZcZZcZcZc

8271621522142

213

222

211

8371631523143

213

232

211

8372632523243

223

232

221

=+++++++=+++++++=+++++++

(7)

therein c, d and e are coefficients that are calculated accordingly to the parameters and generalized coordinates, namely the length limbs of the parallel manipulator.

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Expressions (7) are similar to those introduced in Tsai (1999); however their derivation is simpler due to the inclusion, in this contribution, of Eqs. (2). Please note that only two of the unknowns are present in each one of Eqs. (7) and therefore

their solutions appear to be an easy task. For example, 2Z and 3Z can be obtained as

functions of 1Z from the last two quadratic equations; afterwards the substitution of these

variables into the first quadratic yields a highly non-linear equation in 1Z . The handling of

such an expression is a formidable an unpractical task. Thus, an appropriated strategy is required for solving the system of equations at hand. Some options are • A numerical technique such as the Newton-Raphson method. It is an effective option,

however only one and imperfect solution can be computed, and there are not guarantee that all the solutions will be calculated.

• Using computer algebra like Maple©. An absolutely viable option that guarantee the computation of all the possible solutions.

• The application of the Sylvester dialytic elimination method. An elegant option that allows to compute all the possible solutions.

In this contribution the last option was selected and in what follows the results will be presented.

With the purpose to eliminate 3Z , the first two quadratics of (7) are rewritten as follows

06p3Z5p2

3Z4p

03p3Z2p23Z1p

=++=++

(8)

where 1p , 2p and 3p are second-degree polynomials in 2Z while 4p , 5p and 6p are

second-degree polynomials in 1Z . After a few operations, the term 3Z is eliminated from

(8). With this action, two linear equations in two unknowns, the variable 3Z and the scalar 1,

are obtained. Casting in matrix form such expressions it follows that

⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣

⎡ =0

0

13Z

1M (9)

where

⎥⎦⎤⎢⎣

⎡−−−−=

6p2p5p3p6p1p4p3p4p3p6p1p4p2p5p1p

1M

It is evident that expression (9) is valid if, and only if, ( ) 0Mdet 1 = . Thus clearly one can

obtain

011p2Z10p22Z9p

32Z8p

42Z7p =++++ (10)

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where 7p , 8p , 9p , 10p and 11p are fourth-degree polynomials in 1Z ; and the first step of

the Sylvester dialytic elimination method finishes with the computation of this eliminant. Please note that Eq. (10) and the last quadratic of Eqs. (7) represents a non-linear

system of two equations in the unknowns 1Z and 2Z , and in what follows it is reduced into

an univariate polynomial equation. As an initial step, that last quadratic of (7) is rewritten as

014p2Z13p22Z12p =++ , (11)

where 12p , 13p and 14p are second-degree polynomials in 1Z . It is very tempting to

assume that the non-linear system of two equations formed by (10) and (11) can be easily

solved obtaining first 2Z in terms of 1Z from Eq. (11) and later substituting it into Eq. (10).

However, when one realize this apparent evident action with the aid of computer algebra, an excessively long expression is derived, and its handling is a hazardous task. Thus, the application of the Sylvester dialytic elimination method is a more viable option. In order to avoid extraneous roots, it is strongly advisable the deduction of a minimum of

linear equations. For example, the term 42Z is eliminated multiplying Eq. (10) by 12p and Eq.

(11) by 227Zp . The substraction of the obtained expressions leads to

011p12p2Z10P12P22)Z9p12p7p14(p

32)Z8p12p7p13(p =−−−+− . (12)

Expressions (11) and (12) can be considered as a linear system of two equations in the four

unknowns32Z ,

22Z , 2Z and 1. Therefore it is necessary the search of two additional linear

equations.

An equation is easily obtained multiplying Eq. (11) by 2Z

02Z14p22Z13p

32Z12p =++ . (13)

The search of the fourth equation is more elusive, for details the reader is referred to Tsai

(1999). To this end, multiplicate Eq. (10) by )13212 pZ(p + and Eq. (11) by )228

327 ZpZ(p + .

The subtraction of the resulting expressions leads to

011p13p2)Z10p13p11p12(p

22)Z14p9p9p13p10p12(p

32)Z14p7p9p12(p

=+++−++−

(14)

Casting in matrix form expressions (11)-(14) it follows that

⎥⎥⎥⎦

⎤⎢⎢⎢⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡=

0

0

0

0

12Z

22Z

32Z

2M , (15)

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321

where

⎥⎥⎥⎥⎦

⎤

⎢⎢⎢⎢⎣

⎡

+−+−

−−−−=11p13p10p13p11p12p14p8p9p13p10p12p14p7p9p12p

14p13p12p11p12p10p12p9p12p7p14p8p12p7p13p

14p13p12p

2M0

0

Clearly expression (15) is valid if, and only if, 0) =2det(M . Therefore, this eliminant yields a

sixteenth-order polynomial in the unknown 1Z .

It is worth to mention that expressions (10) and (11) have the same structure of those

derived by Innocenti & Parenti-Castelli (1990) for solving the forward position analysis of

the Stewart platform mechanism. However, this work differs from that contribution in that,

while in this contribution the application of the Sylvester Dialytic elimination method

finishes with the computation of the determinant of a 4x4 matrix, the contribution of

Innocenti & Parenti-Castelli (1990), a more general method than the presented in this

section, finishes with the computation of the determinant of a 6x6 matrix.

Once 1Z is calculated, 2Z and 3Z are calculated, respectively, from expressions (11) and the

second quadratic of (8) while the remaining components of the coordinates, iX and iY , are

computed directly from expressions (5) and (6), respectively. It is important to mention that

in order to determine the feasible values of the coordinates of the points iP , the signs of the

corresponding discriminants of 2Z , 3Z and iY must be taken into proper account. Of

course, one can ignore this last recommendation if the non-linear system (3) is solved by

means of computer algebra like Maple©.

Finally, once the coordinates of the centers of the spherical joints are calculated, the well-

known 44 × transformation matrix T results in

⎥⎦⎤⎢⎣

⎡×

=10

rRT

31

C/O , (16)

where, ( ) 3/321C/O PPPr ++= is the geometric center of the moving platform, and R is the

rotation matrix.

3. Velocity analysis

In this section the velocity analysis of the 3-RPS parallel manipulator is carried out using the

theory of screws which is isomorphic to the Lie algebra e(3). This section applies well

known screw theory; for readers unfamiliar with this mathematical resource, some

appropriated references are provided at the end of this work (Sugimoto, 1987; Rico and

Duffy, 1996; Rico et al, 1999).

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322

The mechanism under study is a spatial mechanism, and therefore the kinematic analysis

requires a six-dimensional Lie algebra. In order to satisfy the dimension of the subspace

spanned by the screw system generated in each limb, the 3-RPS parallel manipulator can be

modelled as a 3-R*RPS parallel manipulator, see Huang and Wang (2000), in which the

revolute joints R* are fictitious kinematic pairs. In this contribution, see Fig. 2, each limb is

modelled as a Cylindrical + Prismatic + Spherical kinematic chain, CPS for brevity. It is

straightforward to demonstrate that this option is simpler than the proposed in Huang and

Wang (2000). Naturally, this model requires that the joint rate associated to the translational

displacement of the cylindrical joint be equal to zero.

Fig. 2. A limb with its infinitesimal screws

Let ),,( ZYX ωωωω = be the angular velocity of the moving platform, with respect to the

fixed platform, and let )v,v,(vv OZOYOXO = be the translational velocity of the point O,

see Fig. 2; where both three-dimensional vectors are expressed in the reference frame XYZ.

Then, the velocity state [ ]OO vωV = , also known as the twist about a screw, of the

moving platform with respect to the fixed platform, can be written, see Sugimoto (1987),

through the j-th limb as follows

O

5

0i

1ij

ij1ii V$ω =∑=

++ { }1,2,3j ∈ , (17)

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323

where, the joint rate jj32 qω $= is the active joint associated to the prismatic joint in the j-th

limb, while 0=j10ω is the joint rate of the prismatic joint associated to the cylindrical joint.

With these considerations in mind, the inverse and forward velocity analyses of the

mechanism under study are easily solved using the theory of screws.

The inverse velocity analysis consists of finding the joint rate velocities of the parallel

manipulator, given the velocity state of the moving platform with respect to the fixed

platform. Accordingly to expression (17), this analysis is solved by means of the expression

O-1jj VJΩ = . (18)

Therein

• [ ]6j

55j

44j

33j

22j

11j

0j $$$$$$J = is the Jacobian of the j-th limb, and

• [ ]Tj65

j54

j43

j32

j21

j10j ωωωωωωΩ = is the matrix of joint velocity rates of the j-

th limb. On the other hand, the forward velocity analysis consists of finding the velocity state of the

moving platform, with respect to the fixed platform, given the active joint rates jq$ . In this

analysis the Klein form of the Lie algebra e (3) plays a central role.

Given two elements [ ]O111 ss$ = and [ ]O222 ss$ = of the Lie algebra e (3), the Klein

form,{ }*,* , is defined as follows

{ } O12O2121 ssss,$$ •+•= . (19)

Furthermore, it is said that the screws 1$ and 2$ are reciprocal if { } 0=21 ,$$ .

Please note that the screw 54

$i is reciprocal to all the screws associated to the revolute joints

in the same limb. Thus, applying the Klein form of the screw 54

$i to both sides of expression

(17), the reduction of terms leads to

{ } i5i

4O q$,V $= { }1,2,3i ∈ . (20)

Following this trend, choosing the screw 65

$i as the cancellator screw it follows that

{ } 0=6i

5O $,V { }1,2,3i ∈ . (21)

Casting in a matrix-vector form expression (20) and (21), the velocity state of the moving

platform is calculated from the expression

QVΔJ OT $= , (22)

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324

wherein

• [ ]63

562

561

553

452

451

4$$$$$$J = is the Jacobian of the parallel manipulator,

• ⎥⎦⎤⎢⎣

⎡×

×=333

333

0I

I0Δ is an operator of polarity, and

• [ ]T321 000qqqQ $$$$ = .

Finally, once the angular velocity of the moving platform and the translational velocity of

the point O are obtained, respectively, as the primal part and the dual part of the velocity

state [ ]OO vωV = , the translational velocity of the center of the moving platform,

vector Cv , is calculated using classical kinematics. Indeed

C/OOC rωvv ×+= . (23)

Naturally, in order to apply Eq. (22) it is imperative that the Jacobian J be invertible.

Otherwise, the parallel manipulator is at a singular configuration, with regards to Eq. (18).

4. Acceleration analysis

Following the trend of Section 3, in this section the acceleration analysis of the parallel

manipulator is carried out by means of the theory of screws.

Let ),,( ZYX ωωωω $$$$ = be the angular acceleration of the moving platform, with respect to the

fixed platform, and let )a,a,(aa OZOYOXO = be the translational acceleration of the point

O; where both three-dimensional vectors are expressed in the reference frame XYZ. Then the

reduced acceleration state [ ]OOO vωaωA ×−= $ , or accelerator for brevity, of the moving

platform with respect to the fixed platform can be written, for details see Rico & Duffy

(1996), through each one of the limbs as follows

Oj-Lie

5

0i

1ij

ij1ii A$ω =+∑=

++ $$ { }1,2,3j ∈ , (24)

where jLie$ − is the Lie screw of the j-th limb, which is calculated as follows

∑= = ⎥⎥⎦⎤

⎢⎢⎣⎡ ++++ ∑+=

4

0k

1rj

rj1rr

1kj

kj1kkj-Lie

5

kr

$ω$ω1

$ ,

and the brackets [ ]** denote the Lie product.

Equation (24) is the basis of the inverse and forward acceleration analyses.

The inverse acceleration analysis, or in other words the computation of the joint acceleration

rates of the parallel manipulator given the accelerator of the moving platform with respect

to the fixed platform, can be calculated, accordingly to expression (24), as follows

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325

)$(AJΩ jLieO-1jj −−=$ , (25)

where [ ]Tj65

j54

j43

j32

j21

j10j ωωωωωωΩ $$$$$$$ = is the matrix of joint acceleration rates.

On the other hand, the forward acceleration analysis, or in other words the computation of

the accelerator of the moving platform with respect to the fixed platform given the active

joint rate accelerations jq$$ of the parallel manipulator; is carried out, applying the Klein form

of the reciprocal screws to Eq. (24), using the expression

QAΔJ OT $$= , (26)

where

{ }{ }{ }{ }{ }{ } ⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−

+++

=

3,

2,

,

3,

2,

,

Lie63

5

Lie62

5

1Lie61

5

Lie53

43

Lie52

42

1Lie51

41

$$

$$

$$

$$q

$$q

$$q

Q$$

$$

$$

$$

Once the accelerator [ ]OOO vωaωA ×−= $ is calculated, the angular acceleration of the

moving platform is obtained as the primal part of OA , whereas the translational acceleration

of the point O is calculated upon the dual part of the accelerator. With these vectors, the

translational acceleration of the center of the moving platform, vector Ca , is computed using

classical kinematics. Indeed

)( C/OC/OOC rωωrωaa ××+×+= $ . (27)

Finally, it is interesting to mention that Eq. (26) does not require the values of the passive

joint acceleration rates of the parallel manipulator.

5. Case study. Numerical example

In order to exemplify the proposed methodology of kinematic analysis, in this section a

numerical example, using SI units, is solved with the aid of computer codes.

The parameters and generalized coordinates of the example are provided in Table 1.

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2πt0

/2)]cos[tsin(t0.35sin(t)q

)]in(t)cos(t0.35sin[tsq

(t)cos(t)2

0.5sinq

aaa

9).963346327 0, 918,(-.2682607u

3).713993824- 0, 970,(-.7001519u

6).249352503- 0, 85,(.96841278u

9).134130395- 0, 640,(-.4816731B

5).350075998- 0, 22,(.35699691B

2).484206394 0, 18,(.12467625B

3

2

1

231312

3

2

1

3

2

1

≤≤−=

=−=

=========

2/3

Table 1. Parameters and instantaneous length of each limb of the parallel manipulator

According with the data provided in Table 1, at the time t=0 the sixteenth polynomial in 1Z

results in

0.=16

1e11Z.261153294

+15

12e12Z.378734907-

14

1e13Z.195532604+

13

1e13Z.373666459-

12

112Z.64783709e

-11

1e13Z.786657045+

10

11Z.3921344e1 +

9

1e13Z.672039554-

8

1e13Z.108993550

-7

15e13Z.273968077

6

1e12Z.964036155+

5

1e12Z.444113311-

4

1e12Z.281160758

-3

110Z.82281001e-

2

1e11Z.246379238+

1e10Z.627748325+09490873788e

+

The solution of this univariate polynomial equation, in combination with expressions (5)

and (6), yields the 16 solutions of the forward position analysis, which are listed in Table 2.

Taking solution 3 of Table 2 as the initial configuration of the parallel manipulator, the most

representative numerical results obtained for the forward velocity and acceleration analyses

are shown in Fig. 3.

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Solution 1P 2P 3P

1,2 .335)- 0.307, (-.086,± .424)- .994,(.432,± .101)- 1.093,(-.364,±

3,4 .471) .899, (.121,± .354)- .999, (.361, ± 0)1.099,-.13(-.468,±

5,6 .625) .888, (.161, ± .231)- .985,(.236,± .151) .273,(.544,±

7,8 .385)- .054,(-.099,± .089) .778,(-.091,± .155) .209,(.558,±

9,10 .857,.749)(.193,± .314) .312,(-.321,± .333,.147)(.528,±

11,12 .869,.709)(.182,± .287,.320)(-.326,± 1)1.056,-.05(-.185,±

13,14 7).194i,-.40(-.104,± .615) .950i,(-.628,± ).004i,.160(.578,±

15,16 7).195i,-.40(-.104,± 4)1.009i,.64(-.657,± .160) .004i,(.578,±

Table 2. The sixteen solution of the forward position analysis

Fig. 3. Forward kinematics of the numerical example using screw theory

Furthermore, the numerical results obtained via screw theory are verified with the help of special software like ADAMS©. A summary of these numerical results is reported in Fig. 4.

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Fig. 4. Forward kinematics of the numerical example using ADAMS©

Finally, please note how the results obtained via the theory of screws are in excellent

agreement with those obtained using ADAMS©.

6. Conclusions

In this work the kinematics, including the acceleration analysis, of 3-RPS parallel

manipulators has been successfully approached by means of screw theory. Firstly, the

forward position analysis was carried out using recursively the Sylvester dialytic

elimination method, such a procedure yields a 16-th polynomial expression in one

unknown, and therefore all the possible solutions of this initial analysis are systematically

calculated. Afterwards, the velocity and acceleration analyses are addressed using screw

theory. To this end, the velocity and reduced acceleration states of the moving platform,

with respect to the fixed platform are written in screw form through each one of the three

limbs of the manipulator. Simple and compact expressions were derived in this contribution

for solving the forward kinematics of the spatial mechanism by taking advantage of the

concept of reciprocal screws via the Klein form of the Lie algebra e (3). The obtained

expressions are simple, compact and can be easily translated into computer codes. Finally, in

order to exemplify the versatility of the chosen methodology, a case study was included in

this work.

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7. Acknowledgements

This work has been supported by Dirección General de Educación Superior Tecnológica, DGEST, of México

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Parallel Manipulators, New DevelopmentsEdited by Jee-Hwan Ryu

ISBN 978-3-902613-20-2Hard cover, 498 pagesPublisher I-Tech Education and PublishingPublished online 01, April, 2008Published in print edition April, 2008

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Parallel manipulators are characterized as having closed-loop kinematic chains. Compared to serialmanipulators, which have open-ended structure, parallel manipulators have many advantages in terms ofaccuracy, rigidity and ability to manipulate heavy loads. Therefore, they have been getting many attentions inastronomy to flight simulators and especially in machine-tool industries.The aim of this book is to provide anoverview of the state-of-art, to present new ideas, original results and practical experiences in parallelmanipulators. This book mainly introduces advanced kinematic and dynamic analysis methods and cuttingedge control technologies for parallel manipulators. Even though this book only contains several samples ofresearch activities on parallel manipulators, I believe this book can give an idea to the reader about what hasbeen done in the field recently, and what kind of open problems are in this area.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

J. Gallardo, H. Orozco, J.M. Rico, C.R. Aguilar and L. Perez (2008). Acceleration Analysis of 3-RPS ParallelManipulators by Means of Screw Theory, Parallel Manipulators, New Developments, Jee-Hwan Ryu (Ed.),ISBN: 978-3-902613-20-2, InTech, Available from:http://www.intechopen.com/books/parallel_manipulators_new_developments/acceleration_analysis_of_3-rps_parallel_manipulators_by_means_of_screw_theory

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